Estimating Copula-Based Extension of Tail Value-at-Risk and Its Application in Insurance Claim

Dependent Tail Value-at-Risk, abbreviated as DTVaR, is a copula-based extension of Tail Value-at-Risk (TVaR). This risk measure is an expectation of a target loss once the loss and its associated loss are above their respective quantiles but bounded above by their respective larger quantiles. In this paper, we propose nonparametric estimators for DTVaR and establish their property of consistency. Moreover, we also propose the variability measure around this expected value truncated by the quantiles, called the Dependent Conditional Tail Variance (DCTV). We use this measure for constructing confidence intervals of the DTVaR. Both parametric and nonparametric approaches for DTVaR estimations are explored. Furthermore, we assess the performance of DTVaR estimations using a proposed backtest based on the DCTV. As for the numerical study, we take an application in the insurance claim amount.

In particular, Chang et al. (2003) introduced three types of VaR nonparametric estimation methods and their corresponding confidence intervals. Brazauskas et al. (2008) and Kaiser and Brazauskas (2006) proposed point and interval estimators for TVaR, as well as proved the consistency of the point estimator. Methni et al. (2014) combined nonparametric kernel methods with extreme-value statistics to find the estimator for TVaR. Dutta and Biswas (2018) compared the performance of nonparametric estimators of TVaR for varying p, namely the empirical estimator, kernel-based estimator, Brazauskas et al.'s estimator, tail-trimmed estimator by Hill, Yamai and Yoshiba's estimator and the filtered historical method. Shen et al. (2019) established empirical likelihood-based estimation with high-order precision for TVaR.
Several extensions of TVaR have also been developed. Jadhav et al. (2013); Wang and Wei (2020); Bairakdar et al. (2020) and Bernard et al. (2020) have modified TVaR by introducing a fixed boundary, instead of infinity, for values beyond the quantile (i.e., VaR). In particular, Jadhav et al. (2013) named the modified risk measure as Modified TVaR (MTVaR). Meanwhile, another extension of TVaR, called Copula TVaR (CTVaR), was suggested by Brahim et al. (2018), in which, they estimate a target loss 1 by involving another dependent or associated loss.
Motivated by the work of Jadhav et al. (2013) and Brahim et al. (2018);  proposed an alternative coherent risk measure that is not only "considering a fixed upper bound of loss beyond VaR" but also "taking into account an associated loss", called Dependent TVaR (DTVaR). Moreover,  proposed an optimization method for DTVaR by applying two metaheuristic algorithms: Spiral Optimization (SpO) and Particle Swarm Optimization (PSO). When we calculate an MTVaR estimate, it will subtract the number of losses beyond VaR and thus make this estimate smaller than the corresponding TVaR. This is a good feature in risk modeling. We argue that this estimate must also be accompanied by an associated risk since this risk scenario occurs in practice; see, for instance, Zhang et al. (2019) and Kang et al. (2019).
The DTVaR can comprehend the connection between bivariate losses and help us to optimally position our investments and enlarge our financial risk protection . In other words, employing the suggested risk measure will enable us to avoid non-essential additional capital allocation while not ignoring other risks associated with the target risk. In this paper, we propose two nonparametric estimators for the risk measure of DTVaR by following the approaches of Brazauskas et al. (2008) and Jadhav et al. (2013). These estimators are proven to be consistent.
Although the DTVaR serves crucial information on the tail distribution of the target loss, the necessity for other risk measures came up in competitive and unpredictable market environments. Principally, realizing that the DTVaR, being the tail mean, is not able to capture the tail variability, we propose a second tail moment or variance in the tail distribution truncated by two pair of VaRs that is called Dependent Conditional Tail Variance (DCTV). This measure can concatenate the dissemination in the tail. Moreover, the DCTV can be considered a generalization of Conditional Tail Variance (CTV) proposed by Furman and Landsman (2006). Using DCTV, we are able to prove the asymptotic normality of DTVaR and even derive confidence intervals for the DTVaR estimators. Just as Righi and Ceretta (2015) used CTV for the backtesting of TVaR estimations using the bootstrap method, we also use DCTV for the backtesting of DTVaR estimations.
The rest of the paper is organized as follows. In Section 2, we briefly explain the novel risk measure of dependent tail VaR. The nonparametric estimation of DTVaR is discussed in Section 3, whereas the truncated variance, called the dependent conditional tail variance, is presented in Section 4. Section 5 presents the parametric estimate of DTVaR in a Pareto case. The choice of the contraction parameters that appear in the definition of DTVaR is considered in Section 6. Conclusions are discussed in Section 7. All mathematical proofs are deferred to Appendix A.

The Dependent Tail Value-at-Risk
Let (Ω, F , P) be an atomless probability space, and L 1 be the set of real integrable random variables (i.e., random variables with finite means) defined on (Ω, F , P). A risk measure is a functional ρ : L 1 → R.
Consider that X and Y are two random losses that are dependent and have marginal distribution functions F X and F Y . Provided a value α ∈ (0, 1), generally close to one, the VaR of X at a probability level α is the quantile Q α of F X for this level. Mathematically, the VaR is defined as follows: Based on this definition, we can note that the VaR does not consider information after the quantile of interest, only the point itself. Moreover, despite its simplicity and ease of implementation, VaR has the shortcoming of not being a coherent risk measure in the sense of Artzner et al. (1999). The TVaR at probability level α is then the expectation of X once X is above the VaR for this level, i.e., an extreme loss. Formally, Formulation (2) defines TVaR.
Note that 1 − α in (2) is the significance level for TVaR.
As we state in Section 1;  proposed another risk measure as a generalization of TVaR that not only considers the information about the potential size of the loss X between two quantiles but also takes into account the excess of another loss Y that is associated with X. Formally, Formulation (3) defines DTVaR.
where Y is another loss that is associated with X (or X depends on Y), Here, α and δ denote the probability level and excess level, respectively. Moreover, X is called the target loss, whereas Y is called the associated loss. In the sequel, two lemmas related to the DTVaR are given.
Lemma 1 . Let X and Y be two random losses with a joint probability function f X,Y . Let α, δ ∈ (0, 1) and a, d ≥ 0 be specified numbers. The Dependent Tail VaR (DTVaR) of X given values beyond its VaR up to a fixed value of losses and a random loss Y is given by In practice, a joint probability function is difficult to find unless a bivariate normal distribution is assumed. For the case of joint exponential distribution, we may refer to Kang et al. (2019) for Sarmanov's bivariate exponential distribution. In most cases, two or more dependent losses rely on a copula in order to have an explicit formula of its joint distribution.
Lemma 2 . Let X and Y be two random losses with a joint distribution function represented by a copula C. Let α, δ ∈ (0, 1) and a, d ≥ 0 be specified numbers. The Dependent Tail VaR (DTVaR) of X given values beyond its VaR up to a fixed value of losses and a random loss Y is given by where F −1 X denotes the quantile function of X, The following property applies to DTVaR. The property states that the DTVaR is a law-invariant convex risk measure. Property 1. The Dependent Tail VaR (DTVaR) is a law-invariant risk measure.

The Estimation of DTVaR
When dealing with real data, it is not always easy for us to know the distribution of the data, even if we use software for fitting distribution. As a result, estimating DTVaR is also not easy. To avoid the difficulty of the parametric estimation of DTVaR, we propose a nonparametric one.
We propose two nonparametric estimators of the DTVaR. The first empirical estimator of DTVaR (δ,d) (α,a) (X|Y; C) is defined as follows. Let (X 1 , Y 1 ), · · · , (X m , Y m ) be a collection of random vectors with size m, where X 1 , · · · , X m and Y 1 , · · · , Y m are independently and identically distributed (iid) random losses, respectively. Suppose that F m,m X,Y denotes the corresponding empirical joint distribution function, which is given by where I(·, ·) denotes the indicator function. In addition, suppose that F m X and F m Y denote the empirical marginal distribution functions of iid X 1 , · · · , X m and iid Y 1 , · · · , Y m ,, which are given by Suppose that F X and F Y denote the unknown distribution functions of X and Y.
Consider a square (0, 1] 2 originating from two intervals (0, 1]. Subdivide each of both intervals (0, 1] into m subintervals ( j−1 m , j m ] and m subintervals ( l−1 m , l m ], j, l = 1, · · · , m, so that we obtain m 2 small squares ( j−1 m , Similarly, when δ ∈ ( l−1 m , l m ] and Note that, in (7), we do not sum Y m(i) but X m(i) paired with Y m(i) . To simplify the notation and computation, we sum X m(i) by applying the indicator function I(y m(l) ≤ Y m(i) ≤ y m(l 1 ) ); thus, we obtain Next, note that where x m(j) and y m(l) , respectively, denote the realizations of X m(j) and Y m(l) , whilst, From (8) and (9), we obtain . Note that, in a similar and simpler way, it can be shown that an estimator of MTVaR (proposed by Jadhav et al. 2013) is given by for all α ∈ ( j−1 m , 1+a . If we adjust the index i in (11) to index q, then MTVaR (α,a) (X) can be rewritten as (see Jadhav et al. 2013) where k(q) = mα (q) , α (q) = α + q(1−α) m(1−α) , q = 0, 1, · · · , (1 − α) 1+a and x denotes the smallest integer that is larger than x.

Following the derivation method of the estimator MTVaR
(1) (α,a) (X) in (12), we obtain where The estimator given in (13) may be improved by considering a smoothed version, which is the second estimator, as follows: In the following theorem, we prove the consistency of DTVaR estimators.

The Dependent Conditional Tail Variance and Confidence Intervals
In addition to the TVaR, some authors also consider the variability of the loss in the tail of the distribution. The notion is that, in spite of its practicality and desired properties, the TVaR only picks up the average loss in the tail and forsakes its variability, and thus it makes sense to concatenate the second tail moment or the variance in the tail distribution. In this regards, Furman and Landsman (2006) put forward the Tail Variance Premiun (TVP) that contains Conditional Tail Variance (CTV), where the last term E[(X − TVaR α (X)) 2 |X ≥ Q α ] is called the CTV. Moreover, Righi and Ceretta (2015) used the square root of the CTV, instead of ordinary variance, for the backtesting of TVaR estimation using the bootstrap method.
Motivated by the CTV proposed by Furman and Landsman (2006), we propose a Dependent Conditional Tail Variance (DCTV) of target loss X associated with another loss Y, Thus, DCTV is a variability or dispersion measure around DTVaR truncated by the VaRs of target loss and associated loss. In addition, since DTVaR is a generalization of TVaR, then DCTV is also a generalization of CTV.
Lemma 3. Let X and Y be two random losses with a joint distribution function represented by a copula C. Let α, δ ∈ (0, 1) and a, d ≥ 0 be specified numbers. The Dependent Conditional Tail Variance (DCTV) of X given values beyond its VaR up to a fixed value of losses and a random loss Y is given by where DTVaR

The Estimation of DCTV
Following the derivation method of the estimators DTVaR  (13) and (14), we obtain two estimators for DCTV and In the following theorem, we prove the consistency of DCTV estimators.

Confidence Intervals for DTVaR
It is obvious that the estimators in (13) and (14) are point estimators for DTVaR. The next step is to construct point-wise confidence intervals for DTVaR. We derive the (pointwise) confidence intervals, whose construction is based on the following asymptotic result.
Theorem 3. Let α, δ ∈ [0, 1] and contraction parameters a and d be fixed. Let the distribution function F X be continuous at the points F −1 X (α) and F −1 X (α 1 ). Then, for n = 1, 2, we have (α,a) (X|Y; C) are given in (5) and (16). In particular, statement (19) holds for any finite contraction parameters a, d ≥ 0 if the distribution function F X is continuous everywhere on the real line.

Parametric Estimation under FGM Copula
In this section, we find parametric estimates for the DTVaR at any given α, δ ∈ (0, 1) and specified contraction parameters a, d ≥ 0 under a Pareto distribution by using the Maximum Likelihood Estimation (MLE) method of Pham et al. (2019). However, before calculating the estimates, we take the following steps: 1.
Derive the expression of the DTVaR for the Pareto distribution; 2.
Calculate the parametric estimates of the distribution parameters of random samples X 1 , · · · , X m and Y 1 , · · · , Y m , each of which is assumed to be a Pareto distribution.
Moreover, we show that the DTVaR, when we consider the correlation (or dependence) between positive quadrant dependent (PQD) losses, is larger than the TVaR. That means, for α, δ ∈ (0, 1), then Note that, in the Negative-Quadrant-Dependent (NQD) losses, we have the reverse of Inequality (21). In particular, also note that 1 − α − δ + C(α, δ; θ) is the joint significance level (j.s.l.) for the DTVaR in (21). We use the j.s.l. for assessing the performance of DTVaR estimation. Now, we derive the expression for the DTVaR for a Pareto distributed loss associated with another loss joined by a Farlie-Gumbel-Morgenstern (FGM) copula that is defined as We are aware that the FGM copula introduces only light dependence. However, it admits positive as well as negative dependence between a set of random variables. The FGM copula is often used in applications to describe dependence structures due to its tractability and simplicity (see, for instance, Bargès et al. 2009;Chadjiconstantinidis and Vrontos 2014;and Jiang and Yang 2016).
Suppose that X is a Pareto random loss with parameter (γ 1 , β 1 ). Suppose also that our dependent (associated) random loss Y is following Pareto distribution with parameter (γ 2 , β 2 ). The distribution function of X and Y are, respectively, F X (x) = 1 − (β 1 /(x + β 1 )) γ 1 for x ≥ 0, and F Y (y) = 1 − (β 2 /(y + β 2 )) γ 2 for y ≥ 0. Their inverses are easy to find and thus their VaRs are as well, which are The risk measure DTVaR formula for X given Y, under the FGM copula, may be found by using Lemma 2.
Lemma 4. Let X and Y be two Pareto distributed random variables with parameters (γ 1 , β 1 ) and with parameter (γ 2 , β 2 ). Suppose that the joint distribution of X and Y are defined by a bivariate FGM copula as follows: Then, the DTVaR of X given Y at levels α and δ, 0 < α, δ < 1, is where θ denotes the dependence (or Copula) parameter between X and Y, the copula C(p, q; The corresponding parametric estimate of DTVaR in (22) is then found by replacing unknown parameters γ 1 , β 1 and θ with their respective estimates. That is, we have Example 1. Let X i , i = 1, 2, 3 have a Pareto distribution with parameters γ i = 3 and β 1 = 2500, i.e., P a(3, 2, 500). Let Y be another Pareto random loss that also has a Pareto distribution and associates with X i . Both Figures 1 and 2 present the DTVaR (δ,0) (0.9,0) estimates for various FGM copula parameters and δ ∈ (0.1, 1), along with the TVaR 0.9 estimates. Consider the bivariate losses (X i , Y), i = 1, 2, 3. For each couple (X i , Y), we set θ 1 = 1, θ 2 = 0.5 and θ 3 = 0.01, respectively (see Figure 1a). The selection of parameters θ i , i = 1, 2, 3 corresponds, respectively, to the strong, medium and weak dependences. In Figure 1a, the comparison of the riskiness of X 1 , X 2 and X 3 is presented. Notice that the risk measures of the TVaR of X i at level α are the same in the three cases. Furthermore, note that DTVaR coincides with TVaR in the independence case (θ = 0), whereas DTVaR is exactly the same as CTVaR when a = d = 0. The DTVaR of the loss X 1 is higher than those of X 2 and X 3 , respectively, i.e., X 1 is riskier than X 2 and X 3 . In Figure 1b, it is shown that both DTVaR and TVaR of X 1 are located above the VaR of X 1 for the same probability level α. We can see that the DTVaR estimates are always larger than the TVaR estimates when the copula parameters are positive, whereas the DTVaR estimates are always smaller than the TVaR estimates when the opposite occurs (see Figure 2). Therefore, these results are in accordance with the statement (21) and its reverse.

Data Analysis
We have used the data of one-year vehicle insurance policies from Macquarie University (2005). This data set is based on one-year vehicle insurance policies taken out in 2004 or 2005. There are 67,856 policies, of which, 4624 (6.8%) had at least one claim. To be clear, the vehicle values written in the source (data set) are values in USD 10,000 s. Out of 4624 policies, there are six observations whose vehicle value is 0. We do not include these six observations in the calculation of DTVaR estimations. Therefore, the data size is m = 4618. Suppose that the target loss X is the insurance claim amount and the associated loss Y is the vehicle value. Table 1 provides summary statistics on the claim amount and vehicle value. We can find that both the claim amount and vehicle value have positive skewness, namely 5.0470 and 1.8614, respectively. Moreover, the respective kurtosis of the claim amount and vehicle value is significant when higher than 3. Kurtosis values above 3 (43.3102 and 9.9344) indicate that, relative to a normal distribution, more probability tends to be at points away from the mean than at points near the mean. This is confirmed by Figure 3. Moreover, Figure 3c shows the box plot of the data of the claim amount, depicting that there are 758 outliers in the right tail.  In this section, we compute the parametric estimates of DTVaR. Before comparing the parametric and nonparametric (empirical) estimates of DTVaR, we compute the estimators of DTVaR suggested in Section 3.
Our empirical analysis supports the claim that the suggested risk measure does not underestimate or overestimate the actual risk for a = 0 and various δ. Thus, DTVaR is quite meaningful. However, the DTVaR does not necessarily estimate the actual risk properly when the contraction parameters a > 0 and d > 0. This is not surprising, because of the presence of the excess level δ (other than the probability level α), together with contraction parameter d, which also contributes to the DTVaR estimation. We employ the hypothesis testing procedure (backtesting) to verify this claim empirically. In the future, we need to concurrently estimate a > 0 and d > 0, which can optimize DTVaR using numerical optimization so that DTVaR estimates the actual risk properly.

Parametric Preliminary Results
By using the MLE method, we obtain the results that X and Y are both Pareto distributed with parameter estimates that areγ 1 = 2.0468,β 1 = 2203.9 for X, andγ a = 295.12, β 1 = 563.44 for Y. In particular, for X, the estimation of parameters is very likely to be influenced by the large number of outliers in the claim amount. Again, by using the MLE method, we obtain the estimate of FGM copula parameterθ FGM = 0.0221.
Consider the bivariate loss (X, Y). For (X, Y), we haveθ FGM = 0.0221. Furthermore, note again that DTVaR coincides with CTVaR when a = d = 0. In Figure 4a, it is evident that both DTVaR (δ,0) (0.9,0) and DTVaR (δ,0) (0.9,0.01) estimates are located between the estimates of VaR with different probability levels. It is interesting to note that DTVaR (δ,0) (0.9,0.01) is smaller than both DTVaR (δ,0) (0.9,0) and TVaR 0.9 (see Figure 4a-c). This fact indicates that DTVaR is much more flexible than TVaR and CTVaR, i.e., DTVaR can be set as equal to or less than CTVaR, or even less than TVaR, by carefully determining the parameters a and d. Furthermore, the results in Figure 4b,c show the same pattern as the results in Figure 1, i.e., the estimates of DTVaR are larger than those of TVaR, which is due to the positive copula parameter estimate.

Backtesting
In the backtesting for the DTVaR, we are interested in the size of the discrepancy between the claims above the VaR estimate and the estimate of the DTVaR when VaRs violation occurs. A VaRs violation occurs when the actual loss is larger than the estimated figures at specified probability and excess levels. These discrepancies can be positive, negative or zero. We assume that these discrepancies (also called residuals) are iid, conditioned on claims that are larger than the VaRs estimates. We propose an adaptation (generalization) of the Righi and Ceretta (2015) procedure. This approach is based on series r, which represents the residual exceedances over the VaR, i.e., the violations standardized by the DTVaR estimate and the DCTV estimate of claim X. Given a probability level α and an excess level δ, we can formally represent r by formulation It is clear that we consider the standard deviation truncated by the VaRs, which is the square root of the presented measure DCTV. Similar to Righi and Ceretta (2015), under the null hypothesis, r has a zero mean, against the alternative that the mean of r is positive or negative. This alternative hypothesis represents the real danger, which is underestimating or overestimating loss. Once there is a violation, we take into consideration the information regarding the quantile that the VaR is calculated rather than all of the distribution. Instead of using the p-value for the hypothesis, we use the confidence interval (CI) at a confidence level 1 − γ, which is calculated based on 1000 bootstrap samples (see Righi andCeretta 2015 andJadhav et al. 2013). We reject the null hypothesis when the resulting bootstrap CI contains 0.

Result Analysis
We have estimated the DTVaR and DCTV based on data of one-year vehicle insurance policies. Figures 5-7 and Tables 2-5 present estimates of the DTVaR and DCTV of the respective probability levels and excess levels for various values of a and d. In Tables 3-5, the abbreviations LCL and UCL denote the lower confidence level and upper confidence level of a CI at confidence level γ = 0.95. Note that, according to CIs (20), we have where DTVaR (δ,d)(n) (α,a) (X|Y; C) are given in (13) and (14), and DCTV (δ,d)(n) (α,a) (X|Y; C) are given in (17) and (18).
In particular, Table 2 shows the number as well as the percentage of violations of DTVaR estimations, i.e., the assessment of accuracy for the DTVaR estimates. The assessment is carried out by first observing the joint significance level. For example, in Table 2 (first row, first column), a 0.95% joint significance level (j.s.l.) is lower than 10%. This means that the DTVaR estimates are quite accurate. In the second place, by calculating the number of violations against the DTVaR (0.9,0)(1) (0.9,0) (X|Y; C), DTVaR (0.9,0)(2) (0.9,0) (X|Y; C) and DTVaR (0.9,0)(p) (0.9,0) (X|Y; C), we obtain the percentages of violations of 1.34%, 1.41% and 2.81%, respectively. The number 1.34% is obtained from the division between 62 and 4618, where 62 is the number of violations and 4618 is the total number of observations. Essentially, the number of violations is the number of observations located outside of the critical value, i.e., greater than the DTVaR estimate. These computations are shown for different α and δ. Note that, for various α and δ, the differences between j.s.l. and the percentage of violations for the parametric estimates are always greater than those for the two nonparametric ones. This result implies that we should look for another distribution that is more fit for the variable of the claim amount. We also obtain the fact that the smaller the excess level δ, the smaller the differences between j.s.l. and the percentage of violations. This implies that both nonparametric estimators accurately estimate the DTVaR at an excess level of δ = 0.9.
In Figure 5, we can see that the estimates of DTVaR are relatively larger than those of TVaR. Those relatively large DTVaR estimates are highly probably influenced by the number of outliers (758 observations). In Figure 6a, it can be seen that both first and second estimates of DTVaR relatively nearly coincide. Furthermore, we can see in Figure 6b,c that both first and second estimates of DTVaR are larger than the corresponding estimates of DCTV.  Table 3, we can see that, for a = d = 0, all bootstrap CIs contain 0. We can see similar results from Table 4, where, for a = 0, d = 0, all bootstrap CIs also contain 0. These results indicate that the null hypothesis cannot be rejected, which supports the suggested estimation method for DTVaR (δ,0) (α,0) (X|Y; C) and DTVaR (δ,d) (α,0) (S N |Y; C), that is, there is no underestimation or overestimation of the target loss (claim amount). From both tables, as the values of probability and excess levels increase, estimates of the DTVaR also increase, which is quite obvious. It is interesting to note that, in Table 4, when δ = 0.92, estimates of DTVaR are greater at d = 0.025 than at d = 0.015, but when δ = 0.98, estimates of DTVaR are smaller at d = 0.025 than at d = 0.015. Table 5 shows different results from Tables 3 and 4. Although we can see that the larger the values of probability and excess levels, the larger the estimates of the DTVaR, it is interesting to note that, for several pairs (a, d), estimates of DTVaR fail the backtest. These results indicate that DTVaR estimation is complicated. To overcome this problem, we suggest in the future that the contraction parameters a and d be determined by performing DTVaR optimization so that DTVaR can properly estimate risk. Note that LCL (1) , LCL (2) , UCL (1) and UCL (2) in Tables 2-5 are calculated using Formulas (25) and (26). Figure 7 presents two different results regarding the difference between the parametric estimates of the DTVaR (0.96,d) (0.96,a) and its nonparametric estimates. When the contraction parameter a = d = 0.01, we can see in Figure 7a that the differences between the two are relatively large, and even very large for δ values approaching 1. However, for a = d = 0.1, the differences between the two estimates are relatively small (see Figure 7b).

Conclusions
In this paper, we study a recent coherent risk measure called Dependent Tail Valueat-Risk (DTVaR) initially proposed by , and suggest the estimators. We have proven the consistency of the estimators. Moreover, we also derive a parametric estimate of DTVaR for Pareto distribution under an FGM copula. For the backtesting of DTVaR estimation, we have also suggested a novel variability measure called Dependent Conditional Tail Variance (DCTV), instead of an ordinary variance of the target loss, along with the estimators for DCTV. Additionally, using DCTV, we establish the asymptotic normality of DTVaR estimators and construct confidence intervals for DTVaR.
We found that the nonparametric estimators are more accurate at estimating DTVaR than the parametric estimator. This result implies that we should look for other distributions that are more fit for the variable of the claim amount. Moreover, we will find the DTVaR formulas of the claim amount for exponential and lognormal distributions since the application of both distributions covers actuarial science. Then, we will again compare the accuracy of the DTVaR parametric estimators for exponential and lognormal distributions to the counterpart nonparametric estimators. In the empirical results, the bootstrap CIs in the backtesting procedure have also confirmed that the estimates of the DTVaR do not underestimate or overestimate the actual loss when a = d = 0 or a = 0. However, the DTVaR does not necessarily estimate the actual risk properly when the contraction parameters a > 0 and d > 0. The limitation in our research is that the data of the claim amount contain a large number of outliers, i.e., 16.41% of all observations. This situation may be the reason for why DTVaR (δ,0) (α,0) estimates are relatively much larger than both DTVaR (δ,d) (α,a) and TVaR estimates. In this case, if the risk measure of DTVaR (δ,0) (α,0) is employed to the insurance company, this will push the company to prepare a very large extra fund, which is not necessary. In the future, we will apply Archimedean copulas, such as Clayton and Gumbel, to DTVaR. Archimedean copulas are broadly used in implementations due to their easy form, a diversity of dependence structures and other "nice" properties (Brahim et al. 2018). For fixed level p = α and specified a and d, the DCTV of X associated with Y is given by We suppose that the densities of F X and F Y are f X and f Y , respectively. Thus, Theorem A1 (Glivenko-Cantelli Theorem). Suppose that X 1 , ..., X m are i.i.d. random variables from a distribution with distribution function F X . For each m, let F m X be the empirical distribution function given by Proof for Theorem 2. Before proving the theorem, we state the Glivenko-Cantelli theorem. The proof is similar to the proof for Theorem 1. The statement of Theorem 2 is almost surely equivalent to the convergence of This latter convergence is followed (even uniformly over all α ∈ [0, 1]) if the statement 1 0 1 0 (F m(−1) X (u)) 2 − (F −1 X (u)) 2 d(u, v) → a.s. 0 (A2) holds. We now provide proof in the following steps: • Step 1. Assuming that the random variables X 1 , · · · , X m ∈ R are i.i.d. with distribution function F X , Brazauskas et al. (2008) argued that the bi-implication-the statement (A3) below- is true if and only if the following two statements F m X ⇒ F X (weak convergence) and |x| dF m X (x) → |x| dF X (x) hold. The first statement follows from the classical Glivenko-Cantelli theorem, which says that the supremum distance between F m X and F X converges almost surely to 0. • Step 2. Similarly to Brazauskas et al. (2008), we argue that the statement (A2) is true if the following two statements (F m X ) 2 ⇒ (F X ) 2 and |x| 2 dF m X (x) → |x| 2 dF X (x) almost surely hold. However, previously, we know the fact that F m X ⇒ F X (weak convergence) and |x| dF m X (x) → |x| dF X (x) almost surely hold from Step 1. Then, we have given in (18), we observe that α (q) → α as m → ∞, and, thus, results in h k(q) ≈ 0. Therefore, DCTV (δ,d)(2) (α,a) ≈ DCTV (δ,d)(1) (α,a) , and thus the consistency property also follows for DCTV (δ,d) (2) (α,a) . This finishes the proof of Theorem 2.
Proof for Theorem 3. In the sequel, we have followed the proof of the asymptotic property of the TVaR estimator, originally given in Brazauskas et al. (2008), to prove the asymptotic property of the DTVaR. We start the proof of Theorem 3 with the representation DTVaR (δ,d)(n) (α,a) (X|Y; C) − DTVaR Our next step is to extract a sum of random variables from the right-hand side of (A4). To understand how to perform this well, we shall now look at the integral below: Note that the integral (A5) can be approximated as follows (compare Brazauskas et al. (2008)): Hence, for every fixed α, δ ∈ (0, 1), as well as a, d ≥ 0, we have that √ where .

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In description we use the terms loss(es) and risk(s) interchangeably.